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When you start factoring polynomials, the first step is often to look for common factors. Common factors are numbers or variables that can evenly divide each term in the polynomial. Identifying these common factors makes the job of factoring much simpler.

Example:
In the polynomial \(3x^{2} + 3x - 36\), each term contains the number 3. This means 3 is a common factor. Factoring out this 3, you simplify the polynomial to \(3(x^{2} + x - 12)\).

Identifying common factors assists in breaking down the polynomial into smaller, more manageable parts. This systematic approach helps in simplifying complex expressions and is an essential first step in polynomial factoring.

A quadratic trinomial is a polynomial with three terms that can often be written in the form \(ax^2 + bx + c\). Specifically, when \(a = 1\), it simplifies factoring because the leading coefficient is 1, creating less complexity in the calculation.

Factoring Example:
Consider the quadratic trinomial \(x^2 + x - 12\). To factor this expression, you need to determine two numbers that multiply to the constant term (-12) and simultaneously add up to the coefficient of \(x\) (which is 1 in this case).

• The two numbers are 4 and -3.
• This means that \(x^2 + x - 12\) can be factored into \((x + 4)(x - 3)\).

Understanding how to factor a quadratic trinomial involves recognizing patterns and using methods like trial and error, making it a fundamental skill in algebra.

Once you have identified all the parts and successfully factored each, you obtain what is known as the factored expression. A factored expression is the simplest form of the polynomial, showing all its factors.

Combining Factors:
From the original polynomial \(3x^{2} + 3x - 36\), you've factored out the common factor to achieve \(3(x^2 + x - 12)\). Then, you further factor \(x^2 + x - 12\) into \((x + 4)(x - 3)\).

Putting this all together, the complete factored expression becomes \(3(x - 3)(x + 4)\). This expression is not only simplified but also provides valuable insights into the roots and solutions of the original polynomial.

• It reveals the zeros of the polynomial at \(x = 3\) and \(x = -4\).
• It helps in understanding the behavior of the polynomial function.

Mastering the concept of creating a factored expression allows deeper insights into solving polynomial equations effectively.